Modeling Infrastructure and Network Industries: Theory and - - PowerPoint PPT Presentation

Modeling Infrastructure and Network Industries: Theory and Applications*

Steven A. Gabriel Steven A. Gabriel

Project Management Program, Dept. of Civil & Project Management Program, Dept. of Civil & Env

• Env. Engineering, University of Maryland,

. Engineering, University of Maryland, College Park Maryland, 20742 USA College Park Maryland, 20742 USA Applied Mathematics and Scientific Computation Program, Universi Applied Mathematics and Scientific Computation Program, University of Maryland, College ty of Maryland, College Park, Maryland 20742 USA Park, Maryland 20742 USA Gilbert F. White Fellow, Resources for the Future, Washington, D Gilbert F. White Fellow, Resources for the Future, Washington, DC USA (2007 C USA (2007-

• 2008)

2008) Visiting Scholar, LMI Research Institute, McLean, Virginia, USA Visiting Scholar, LMI Research Institute, McLean, Virginia, USA (2007 (2007-

• 2008)

2008)

Presented at Presented at Infraday Infraday 2007 2007 Berlin, Germany Berlin, Germany October 6, 2007 October 6, 2007

*National Science Foundation Funding, Division of Mathematical S *National Science Foundation Funding, Division of Mathematical Sciences, Awards ciences, Awards 0106880, 0408943

2 2

Outline of Presentation

Briefly, My Background From Optimization to Complementarity Problems then

• n to MPECs and EPECs: Why All the Fuss?

– Complementarity Problem Application: Natural Gas Market Equilibrium

Stochastic Optimization Models

– Stochastic Multiobjective Optimization Application: Telecommunications Network Reconfiguration

Conclusions and Future Work General invitation to Trans-Atlantic Critical Infrastructure Modeling Conference at Univ. of Maryland, Nov. 2, 2007

3 3

My Background

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University of Maryland My Affiliations

– Department of Civil & Environmental Engineering – Applied Mathematics and Scientific Computation Program – Engineering and Public Policy Program (joint between Engineering School and Public Policy School)

5 5

Overview of Research

Research: Main Topics

– Mathematical modeling in engineering-economic systems usually involving critical infrastructure using optimization and equilibrium analysis

• energy market models (natural gas and electricity)
• transportation/traffic
• land development (Multiobjective optimization for “Smart Growth”

in land development)

• wastewater treatment (Optimization and statistical modeling in

biosolids)

• telecommunications (Optimization)

– Development of algorithms for solving equilibria in energy & transportation systems and other planning problems – Development of general purpose algorithms for equilibrium models (using the nonlinear complementarity format)

6 6

Overview of Research

Design of Optimization/ Complementarity Algorithms Analysis

• f Public

Policy Issues Mathematical Modeling

• f

Critical Infrastructure

7 7

From Optimization to Complementarity Problems then on to MPECs and EPECs: Why All the Fuss?

8 8

Example of an Equilibrium Problem A Variation on a Transportation Problem

10 5 4 6 2 10

1 S1=20 S2=20 2

Supplies

1 2 3 D1=10 D2=10 D3=10

Demands

i j

cij

9 9

Example of an Equilibrium Problem A Variation on a Transportation Problem

10 10 10

1 S1=20 S2=20 2

Supplies

1 2 3 D1=10 D2=10 D3=10

Demands

1 =

ψ 3

2 =

ψ 9

1 =

θ 5

2 =

θ 4

3 =

θ

Solution:

• flow on arcs
• dual prices at

nodes

10 10

ij ij 1 13 3 13 2 23 3 23

Optimality conditions are of the form c , 1,2, 1,2,3 +c ,(+ other conditions) Example: +c 4 4 and 10 +c 3 10 4 and

i j ij i j

i j x x x ψ θ ψ θ ψ θ ψ θ + ≥ = = > ⇒ = = + ≥ = > = + > = =

Example of an Equilibrium Problem A Variation on a Transportation Problem

11 11

Example of an Equilibrium Problem A Variation on a Transportation Problem

demand and supply dependent

• price

using before stated conditions

• ptimality

the generalize Can 2. why? demand) for 3 , 2 , 1 j , supply, for 1,2 i , ( prices e appropriat the

• f

function a as vary to them allowing than realistic less is this constants, as given were quantities demand and supply The . 1 : marks Re

j i

= θ = ψ

12 12

Example of an Equilibrium Problem A Variation on a Transportation Problem

( ) ( ) ( ) ( ) ( )

3 3 3 2 2 2 1 1 1 2 2 2 1 1 1

D 14 D D 5 . 10 D D 19 D Demand 1 S 2 . S 20 S S Supply : functions demand and supply (inverse) following the Assume − = θ − = θ − = θ − = ψ − = ψ

( )

( )

j j i i

D S θ ψ

i

S

j

D

( )

i i S

ψ

( )

j j D

θ

13 13

Example of an Equilibrium Problem A Variation on a Transportation Problem

( )

( )

( )

( )

ij ij 3 1 2 1

Complete Optimality Conditions c , 0, 1,2, 1,2,3 c , 1,2 , 1,2,3 This is an example of a complementarity problem (Spatial Price Equilibrium)

i i j j ij ij i i j j i ij j j ij i

S D x i j x S D S x i D x j ψ θ ψ θ

= =

+ ≥ ≥ = = > ⇒ + = = = = =

∑ ∑

14 14

NLP QP LP Complementarity Problems vis-à-vis Optimization and Game Theory Problems

Other non-optimization based problems e.g., spatial price equilibria, traffic equilibria, Nash- Cournot games, zero- finding problems

Complementarity Problems

15 15

Complementarity Problems and Variational Inequalities

Complementarity Problems Variational Inequality Problems But, when polyhedral constraints, VI is a special case of the mixed complementarity problem

16 16

Optimization vs. Complementarity Problems Complementarity problems are more general covering:

– Zero-finding problems – Optimization problems (via Karush-Kuhn-Tucker conditions) – Game Theory problems (e.g., Bimatrix or Nash-Cournot games) – Host of other interesting problems in engineering and economics

Thus, theorems and algorithms designed for CPs can be applied to a wide variety of applications Some problems have no natural optimization counterpart (e.g., via Principle of Symmetry), therefore, can only use CPs in this context CPs very useful for solving policy-related network infrastructure problems (cf. SPE)

– Can include some network participants having market power – Can include other players as price-takers

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Optimization vs. Complementarity Problems (con’t) Complementarity problems can also include problems in which prices (Lagrange multipliers) appear in the primal formulation

– PIES energy infrastructure model of the 1970s – More generally infrastructure models whose modules might represent a detailed sector (e.g., power production) and for which subsets of prices and quantities (and other variables) are passed between these modules, e.g., National Energy Modeling System

Source: http:// Source: http://enduse.lbl.gov/Projects/NEMS.gif enduse.lbl.gov/Projects/NEMS.gif

• S. A. Gabriel, A. S. Kydes, P. Whitman, 2001.

"The National Energy Modeling System: A Large-Scale Energy-Economic Equilibrium Model," Operations Research, 49 (1), 14-25.

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Extensions of Complementarity Problems: MPECs and EPECs Stackelberg Games or More Generally MPECs

– What if two-level problem where top level is a dominant company or the government and bottom level is the rest of the market – This is no longer a complementarity problem since all the players are not at the same level – Instead it’s an example of a mathematical program with equilibrium constraints (MPEC)

19 19

Extensions of Complementarity Problems: MPECs and EPECs Stackelberg Games or More Generally MPECs

– x upper-level planning variables, y lower-level variables, S(x) solution set of lower-level problem (e.g., Nash-Cournot game or optimization) – Lately a number of research papers on MPECs in energy infrastructure planning, transportation planning, etc.

min ( , ) . . ( ) f x y s t x y S x ∈Ω ∈

20 20

Extensions of Complementarity Problems: MPECs and EPECs EPECs

– Can also make the top level a game to get equilibrium problems with equilibriuim constraints (EPEC)

MPECs and EPECs are hard problems for several reasons

– Feasible region not generally known in closed form (can use KKT conditions though) – Instance of global optimization problem

Advantages for regulators

– Can more accurately reflect market behaviors when both strategic players exist in combination with non-strategic ones – Can allow regulators to see what effects for certain potential regulations

• r policies might be on the market with better feedback mechanisms

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Example of Complementarity Problem for Natural Gas Infrastructure Planning

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The Natural Gas Supply Chain

INDUSTRIAL CITY GATE STATION

COMMERCIAL

RESIDENTIAL

DISTRIBUTION SYSTEM UNDERGROUND STORAGE TRANSMISSION SYSTEM Cleaner Compressor Station GAS PROCESSING PLANT GAS PRODUCTION Gas Well Associated Gas and Oil Well

Impurities Gaseous Products Liquid Products

ELECTRIC POWER

From well-head to burner-tip

23 23

Recent Complementarity Modeling and Natural Gas Markets: Gabriel et al.

North America Natural Gas Markets

1. S.A. Gabriel, S. Kiet and J. Zhuang (2005), A Mixed Complementarity-Based Equilibrium Model of Natural Gas Markets, Operations Research, 53(5), 799-818. 2. S.A. Gabriel, J. Zhuang and S. Kiet (2005), A Large-Scale Complementarity Model of the North American Natural Gas Market, Energy Economics, 27, 639-665. 3. S.A. Gabriel, J. Zhuang and S. Kiet (2004), A Nash-Cournot Model for the North American Natural Gas Market, IAEE Conference Proceedings, Zurich, Switzerland, September.

European Union Natural Gas Markets

1.

• R. Egging and S.A. Gabriel (2006), Examining Market Power in the European Natural Gas

Market, Energy Policy, 34 (17), 2762-2778. 2.

• R. Egging, S.A. Gabriel, F.Holz, J. Zhuang, A Complementarity Model for the European

Natural Gas Market, November 2006, in review.

General Natural Gas Markets and Algorithms

1. S.A. Gabriel and Y. Smeers (2006), Complemenatarity Problems in Restructured Natural Gas Markets, Recent Advances in Optimization. Lecture Notes in Economics and Mathematical Systems, Edited by A. Seeger,Vol. 563, Springer-Verlag Berlin Heidelberg, 343-373. 2.

• J. Zhuang and S.A. Gabriel (2006), A Complementarity Model for Solving Stochastic Natural

Gas Market Equilibria Energy Economics, in press. 3. S.A. Gabriel, J. Zhuang, R. Egging, Solving Stochastic Complementarity Problems in Energy Market Modeling Using Scenario Reduction, November 2006, in review.

24 24

Global Aspects of Natural Gas Markets

• Previously, natural gas was more of a continental market

– Pipeline access issues – Market structures

• Now more or less a global market

– Importance of natural gas for more environmentally-friendly power generation – Greater activity in LNG transport – Market restructuring (at least in the US and the EU)

• Main result is that there is a “domino” effect relative to supply security

– Supplier in one country cuts back production or transportation of natural gas – This effects downstream customers who then need more gas from a second supply source – The customers who rely on the second supply source also affected, etc.

• Issues of geopolitical market power being exerted

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Description of Complementarity Model for Global Natural Gas Markets (R. Egging, S.A. Gabriel, F.Holz, J. Zhuang, "A Complementarity

Model for the European Natural Gas Market," November 2006)

Players

– Producers – Traders (marketing aspects of production companies) – Pipeline operators – Storage operators – Marketers – LNG Liquefiers – LNG Regasifiers – Consumers

Multiple seasons Traders (e.g., producers) allowed to have varying degrees

• f market power

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Overall picture

T11 C1 K1,2,3 S1 M1 C3 K1,2,3 S3 M3 R3 L1 Producer Trader Sectors Marketer LNG Liquef Storage LNG Regasif

Country 1 Country 3 Country 2

T31 T31 T32 T12 T13

• Traders are “producer specific contract agents”
• Marketers and storage operators can by from any traders
• Liquefier only buys from transmitter from domestic producer

27 27

Complementarity Aspects Take major players’ economic behavior consistent with maximizing net profit subject to economic and engineering constraints Collect all the resulting optimality conditions along with market-clearing ones Resulting set of conditions is a nonlinear complementarity problem (variational inequality)

28 28

Maximize production revenues less production costs s.t.

– bounds on production rates – bounds on volume of gas produced in time-window of analysis

Decision Variables

– How much to produce in season and year (cubic meters/day)

Market Clearing

– Producers’ sales must equal Trader’s purchases from Producer

Producer’s Problem

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How Will Cutting of Gas from Russia to Ukraine Affect Other Countries? Consider the realistic “domino” effect from one of our recent models (details on model later) Ukraine Disruption Scenario from what actually happened How does Japan get affected?

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Analysis of Strategic LNG Flows From Model

Basic Map from: www.insectzoo.msstate.edu/ Curriculum/ Activities/ WorldMap.html

1 3 .1

LNG modeled as “spot market” Flows in BCM/year

5 5 .5 6 .5 Japan: 75.1

SEA ARB NIG T&T ALG EU UKR RUS JAP AUS KOR US

31 31

Disruption Ukrainian Pipeline

+ 0 .1

Changes in LNG flows. (BCM/year)

• 3 .9

Japan: -3.8 to 71.3

+ 4 .3

• 4 .3

+ 3 .9

• 0 .9

+ 0 .9

• 1 .9

ALG pipes more to EUR, less LNG to USA

T&T EU ALG NIG JAP KOR SEA AUS UKR RUS ARB US

32 32

Stochastic Optimization Models

33 33

Stochastic Optimization Background

( ) ( )

( ) s.t 0, 1, , 0, 1, , : : :

i j n n i n j

Min f x g x i m h x j n f R R g R R h R R ≤ = … = = … → → →

Nonlinear programming problem, objective and constraint functions usually assumed deterministic What if some aspects of these functions (e.g., coefficients) are not known with certainty? This is then a stochastic (nonlinear) programming problem

34 34

Stochastic Optimization Background

• Many method to solve such a stochastic problem, some examples of

approaches – Decomposing the problem (e.g., L-shaped method) – Using a sampling approach – Using a scenario tree for the finite (but usually large) number of realizations, then approximating it with a reduced tree

Römisch, Dupačová, Gröwe-Kuska, Heitsch (2003)

35 35

Stochastic Optimization Background

(J.R. Birge and F. Louveaux, “Introduction to Stochastic Programming,” Springer, New York, 1996)

Stochastic Optimization allows for endogenous handling of risk This is NOT the same as running a number of different scenarios, why? Two important notions: – Expected Value of Perfect Information (EVPI) – Value of Stochastic Solution

36 36

Stochastic Optimization Application: Autonomous Near- Real Time Reconfiguration in Telecommunications Networks

37 37

Free Space Optical Communications Emerging Communications Technology

– A high-speed bridging technology to current fiber optics network – A valuable technology in commercial and military backbone network

Advantage of FSO communications

– Optical wireless (no fibers) – Directional (no frequency interference) – High-speed data rate (~Gbps)

1.25Gbps Optical Transceiver (Canobeam DT-130-LX)

38 38

Topology Control

• Main challenge in FSO networking

– Autonomous Physical Reconfiguration: Pointing, Acquisition, and Tracking – Autonomous Logical Reconfiguration: Topology Optimization

• For example, in the event of a hurricane wiping out links
• Need to reconfigure network quickly
• Challenge

– Responding quickly to a sudden change in link or traffic demand to provide robust quality of service

• Research questions

– How to steer narrow laser beams between two remote optical transceivers automatically and precisely ? – Autonomous Physical Reconfiguration – How to get the optimal topology with respect to physical layer cost or network layer congestion in near-real time? – Autonomous Logical Reconfiguration

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Autonomous Reconfiguration

1 2 4 3

Free Space Optical Network

2 2 3 4 1 1 4 3 3 2 2 3 4 4 1 1

= FSO transceiver

1 2 3 4 =

=

sudden change in traffic demand or link loss

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Generating Cost Matrices: Cloud Model (courtesy: Jaime Llorca, Univ. of Maryland)

41 41

Complexity of Autonomous Logical Reconfiguration

• Candidate solutions are all the

permutations (topologies) of a set of nodes N={1,…,n} – Number of possible topologies = (n- 1)!/2 – e.g., n=12→20 million topologies, n=14 → 3 billion topologies

• For each topology,

– Number of OD pairs = n(n-1) – For each OD pair,

• Two possible routings (clockwise
• r counter-clockwise)
• Number of possible routings =

2^n(n-1) (e.g. n=12→5.4×1039, n=14→6.1×1054)

• This complexity makes it hard to get an
• ptimal topology in real-time.

1 7 5 3 8 2 4 6

destination node

• rigin

node

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Stochastic Multiobjective Optimization Problem (SMOP)

• Jointly minimize cost and congestion to obtain Pareto optimal

topologies

• Objective Function

– Link cost – Uncertainty in traffic demands

• K: # of scenarios (i.e. number of possible traffic demands)
• pk: probability in the realization of the k-th scenario

– Constant weight on each single objective function

• Weight = user’s preference to cost or congestion

1 , 2 ( , ) ( , ) ( , ) 1

min (1 ) ( )

K k k k y f ij ij ij i j

• d

i j k

w c y w p r f SP

ω ω ω= =

+ −

∑ ∑ ∑∑

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SMOP Formulation

bi-connectivity constraints directional flow constraints binary link variable binary flow variable

44 44

Traffic Matrices: Examples

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SMOP Swapping Heuristic: Near-Pareto Optimum (n=10)

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Significant Advantage of MOP Heuristic

• Number of nodes = 20
• Number of weights = 39, i.e. w∈{0.025, 0.05, 0.75, …, 0.975}
• Number of traffic demand scenarios, K = 10, w/

p1=p2=…=p10=1/K=0.1

• MATLAB heuristic code = 39 points × 5 minutes/point = 195 min
• Expected enumeration time = 9,767,520 days (3.6GHz Intel P4)

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Numerical Results: SMOP for n=20

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Conclusions

• Infrastructure planning involves simultaneous consideration of many

important factors, for example:

– Market participants – Regulated/unregulated aspects – Engineering and economic constraints – Benefits to society – Uncertainty in key elements (e.g., demand, weather)

• Need for sophisticated models to take all these factors into account and

provide regulators with accurate tradeoffs for example between

– Level of infrastructure investments – Incentives for socially beneficial directions – Acceptable levels of risk that the infrastructure will be degraded

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Future Work

Continue development of sophisticated engineering- economic models taking into account

– Strategic behavior – Mixture of regulated/unregulated elements – Endogenous treatment of uncertainty

Resulting models will be:

– Stochastic complementarity problems (several efforts on-going now) – Stochastic MPECs – Etc.

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Trans-Atlantic Infraday Conference

General invitation to Trans-Atlantic Critical Infrastructure Modeling Conference at Univ. of Maryland, Nov. 2, 2007 Jointly hosted with German colleagues from Berlin and Dresden Focus on modeling and policy for networked industries: energy, transportation, telecommunications, water Website: http://tai.ee2.biz/

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Back-Up Slides Natural Gas Market Equilibrium Problem

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Major natural gas trade movements (BP Stats)

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j) (i, arc along flow the is x where 3 , 2 , 1 j , 2 , 1 i , x 10 x x 10 x x 10 x x 20 x x x 20 x x x . t . s 10x 2x 6x 4x 5x 0x 1 min program ation) (transport linear following the

• lve

S

ij ij 23 13 22 12 21 11 23 22 21 13 12 11 23 22 21 13 12 11

= = ∀ ≥ = + = + = + ≤ + + ≤ + + + + + + +

Example of an Equilibrium Problem A Variation on a Transportation Problem (Harker)

54 54

Equilibrium Problems: Pure Nonlinear Complementarity Problem

( ) ( ) ( ) ( ) ( )

? equivalent forms two these are why x x F , x , x F form in vector

• r

tarity) (complemen i x F x iii. i x ii. i, x F i. s.t. R an x find R R : F function a Having NCP(F) Problem arity Complement Nonlinear Pure

T i i i i n n n

= ≥ ≥ ∀ = ∀ ≥ ∀ ≥ ∈ →

55 55

Equilibrium Problems: Mixed Nonlinear Complementarity Problem

( ) { } ( ) ( )

n n n i i i i

Mixed Nonlinear Complementarity Problem MNCP(F,l,u) Having a function F:R , respectively, lower, upper bound vectors l,u R

• ,+

with l u find an x R s.t. i

• i. x

F x 0 , ii. x F x ii

n i i i

R and l l u → ∈ ∞ ∞ < ∈ ∀ = ⇒ ≥ < < ⇒ = U

( )

i i

• i. x

F x

i

u = ⇒ ≤

56 56

Max revenues from sales minus production costs, s.t. constraints for production rate and production ceiling are met, and nonnegativity

Producer’s Problem

57 57

Producer’s Karush-Kuhn-Tucker (KKT) Conditions

Marginal Producer Profit

– Marginal profit complementary to sales:

if sales > 0, marginal profit = 0

Marginal Production Rate

• If sales < PR, dual of prod cap a = 0

Production Ceiling

• If tot sales < PROD, dual of prod ceiling b = 0

58 58

Market Clearing in the Production Market

Producer sales must equal purchases by it’s marketing arm & it’s liquefaction plants.

59 59

Back-Up Slides Stochastic Multiobjective Telecommunications Planning Problem

60 60

Numerical Results: Optimum vs. MOP Heuristic Solution (I)

61 61

Numerical Results: Optimum vs. MOP Heuristic Solution (II)

62 62

Numerical Results: Optimum vs. MOP Heuristic Solution (III)

63 63

Numerical Results: Optimum vs. MOP Heuristic Solution (I)

64 64

Cost Matrices: Examples